On the validity of Akaike’s identity for random fields

For univariate stationary and centered time series (Xt)t∈Z, Akaike’s identity links the inverse of the Yule–Walker matrix Γ(p)=E(XX′), where X=(Xt−1,…,Xt−p)′, to the corresponding finite predictor coefficients. It reads as a Cholesky-type factorization Γ(p)−1=L(p)′Σ(p)−1L(p), where L(p) is lower-triangular and Σ(p)−1 is diagonal. Whereas this Cholesky-type factorization exists whenever Γ(p) is positive definite, Akaike derived a meaningful interpretation of L(p) and Σ(p)−1 in terms of finite predictor coefficients. It is useful in many applications and is particularly crucial to derive asymptotic theory for Berk’s spectral density estimator. We investigate the validity of a bona fide extension of Akaike’s identity to univariate stationary random fields (Xt̲)t̲∈Zd. An analogue of Akaike’s result is shown to hold true if and only if the corresponding Yule–Walker matrix Γ(p) is Toeplitz. This condition turns out to be very restrictive and rules out commonly used unilateral autoregressive models in Zd (such as half-plane or quarter-plane fits in Z2). Instead, we prove that the corresponding Cholesky-type factorization Γ(p)−1=L(p)′Σ(p)−1L(p) for random fields does establish a link to a different, but practically less relevant sequence of so-called reshaping past projection coefficients.